Optimal. Leaf size=448 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) \sqrt {d+e x}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)} \]
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Rubi [A] time = 0.20, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} -\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) \sqrt {d+e x}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^{5/2}}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^{7/2}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^{7/2}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{5/2}}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^{3/2}}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 \sqrt {d+e x}}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) \sqrt {d+e x}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{3/2}}{e^6}+\frac {b^{10} B (d+e x)^{5/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{5/2}}+\frac {2 (b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}-\frac {20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {10 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 239, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-21 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+175 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-1050 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)-525 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)+35 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)-21 (b d-a e)^5 (B d-A e)+15 b^5 B (d+e x)^6\right )}{105 e^7 (a+b x) (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 33.34, size = 812, normalized size = 1.81 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (-21 b^5 B d^6+21 A b^5 e d^5+105 a b^4 B e d^5+210 b^5 B (d+e x) d^5-105 a A b^4 e^2 d^4-210 a^2 b^3 B e^2 d^4-1575 b^5 B (d+e x)^2 d^4-175 A b^5 e (d+e x) d^4-875 a b^4 B e (d+e x) d^4+210 a^2 A b^3 e^3 d^3+210 a^3 b^2 B e^3 d^3-2100 b^5 B (d+e x)^3 d^3+1050 A b^5 e (d+e x)^2 d^3+5250 a b^4 B e (d+e x)^2 d^3+700 a A b^4 e^2 (d+e x) d^3+1400 a^2 b^3 B e^2 (d+e x) d^3-210 a^3 A b^2 e^4 d^2-105 a^4 b B e^4 d^2+525 b^5 B (d+e x)^4 d^2+1050 A b^5 e (d+e x)^3 d^2+5250 a b^4 B e (d+e x)^3 d^2-3150 a A b^4 e^2 (d+e x)^2 d^2-6300 a^2 b^3 B e^2 (d+e x)^2 d^2-1050 a^2 A b^3 e^3 (d+e x) d^2-1050 a^3 b^2 B e^3 (d+e x) d^2+105 a^4 A b e^5 d+21 a^5 B e^5 d-126 b^5 B (d+e x)^5 d-175 A b^5 e (d+e x)^4 d-875 a b^4 B e (d+e x)^4 d-2100 a A b^4 e^2 (d+e x)^3 d-4200 a^2 b^3 B e^2 (d+e x)^3 d+3150 a^2 A b^3 e^3 (d+e x)^2 d+3150 a^3 b^2 B e^3 (d+e x)^2 d+700 a^3 A b^2 e^4 (d+e x) d+350 a^4 b B e^4 (d+e x) d-21 a^5 A e^6+15 b^5 B (d+e x)^6+21 A b^5 e (d+e x)^5+105 a b^4 B e (d+e x)^5+175 a A b^4 e^2 (d+e x)^4+350 a^2 b^3 B e^2 (d+e x)^4+1050 a^2 A b^3 e^3 (d+e x)^3+1050 a^3 b^2 B e^3 (d+e x)^3-1050 a^3 A b^2 e^4 (d+e x)^2-525 a^4 b B e^4 (d+e x)^2-175 a^4 A b e^5 (d+e x)-35 a^5 B e^5 (d+e x)\right )}{105 e^6 (d+e x)^{5/2} (a e+b x e)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 592, normalized size = 1.32 \begin {gather*} \frac {2 \, {\left (15 \, B b^{5} e^{6} x^{6} - 3072 \, B b^{5} d^{6} - 21 \, A a^{5} e^{6} + 1792 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 4480 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 3360 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 280 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 14 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 3 \, {\left (12 \, B b^{5} d e^{5} - 7 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \, {\left (24 \, B b^{5} d^{2} e^{4} - 14 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 35 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 10 \, {\left (96 \, B b^{5} d^{3} e^{3} - 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 140 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} - 15 \, {\left (384 \, B b^{5} d^{4} e^{2} - 224 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 560 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 420 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 35 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 5 \, {\left (1536 \, B b^{5} d^{5} e - 896 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 2240 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 1680 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 1101, normalized size = 2.46
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 689, normalized size = 1.54 \begin {gather*} -\frac {2 \left (-15 B \,b^{5} e^{6} x^{6}-21 A \,b^{5} e^{6} x^{5}-105 B a \,b^{4} e^{6} x^{5}+36 B \,b^{5} d \,e^{5} x^{5}-175 A a \,b^{4} e^{6} x^{4}+70 A \,b^{5} d \,e^{5} x^{4}-350 B \,a^{2} b^{3} e^{6} x^{4}+350 B a \,b^{4} d \,e^{5} x^{4}-120 B \,b^{5} d^{2} e^{4} x^{4}-1050 A \,a^{2} b^{3} e^{6} x^{3}+1400 A a \,b^{4} d \,e^{5} x^{3}-560 A \,b^{5} d^{2} e^{4} x^{3}-1050 B \,a^{3} b^{2} e^{6} x^{3}+2800 B \,a^{2} b^{3} d \,e^{5} x^{3}-2800 B a \,b^{4} d^{2} e^{4} x^{3}+960 B \,b^{5} d^{3} e^{3} x^{3}+1050 A \,a^{3} b^{2} e^{6} x^{2}-6300 A \,a^{2} b^{3} d \,e^{5} x^{2}+8400 A a \,b^{4} d^{2} e^{4} x^{2}-3360 A \,b^{5} d^{3} e^{3} x^{2}+525 B \,a^{4} b \,e^{6} x^{2}-6300 B \,a^{3} b^{2} d \,e^{5} x^{2}+16800 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-16800 B a \,b^{4} d^{3} e^{3} x^{2}+5760 B \,b^{5} d^{4} e^{2} x^{2}+175 A \,a^{4} b \,e^{6} x +1400 A \,a^{3} b^{2} d \,e^{5} x -8400 A \,a^{2} b^{3} d^{2} e^{4} x +11200 A a \,b^{4} d^{3} e^{3} x -4480 A \,b^{5} d^{4} e^{2} x +35 B \,a^{5} e^{6} x +700 B \,a^{4} b d \,e^{5} x -8400 B \,a^{3} b^{2} d^{2} e^{4} x +22400 B \,a^{2} b^{3} d^{3} e^{3} x -22400 B a \,b^{4} d^{4} e^{2} x +7680 B \,b^{5} d^{5} e x +21 A \,a^{5} e^{6}+70 A \,a^{4} b d \,e^{5}+560 A \,a^{3} b^{2} d^{2} e^{4}-3360 A \,a^{2} b^{3} d^{3} e^{3}+4480 A a \,b^{4} d^{4} e^{2}-1792 A \,b^{5} d^{5} e +14 B \,a^{5} d \,e^{5}+280 B \,a^{4} b \,d^{2} e^{4}-3360 B \,a^{3} b^{2} d^{3} e^{3}+8960 B \,a^{2} b^{3} d^{4} e^{2}-8960 B a \,b^{4} d^{5} e +3072 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{105 \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right )^{5} e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 647, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \, {\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} A}{15 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (15 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 8960 \, a b^{4} d^{5} e - 8960 \, a^{2} b^{3} d^{4} e^{2} + 3360 \, a^{3} b^{2} d^{3} e^{3} - 280 \, a^{4} b d^{2} e^{4} - 14 \, a^{5} d e^{5} - 3 \, {\left (12 \, b^{5} d e^{5} - 35 \, a b^{4} e^{6}\right )} x^{5} + 10 \, {\left (12 \, b^{5} d^{2} e^{4} - 35 \, a b^{4} d e^{5} + 35 \, a^{2} b^{3} e^{6}\right )} x^{4} - 10 \, {\left (96 \, b^{5} d^{3} e^{3} - 280 \, a b^{4} d^{2} e^{4} + 280 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} - 15 \, {\left (384 \, b^{5} d^{4} e^{2} - 1120 \, a b^{4} d^{3} e^{3} + 1120 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 35 \, a^{4} b e^{6}\right )} x^{2} - 5 \, {\left (1536 \, b^{5} d^{5} e - 4480 \, a b^{4} d^{4} e^{2} + 4480 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 140 \, a^{4} b d e^{5} + 7 \, a^{5} e^{6}\right )} x\right )} B}{105 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt {e x + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.95, size = 718, normalized size = 1.60 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {x^3\,\left (20\,B\,a^3\,b^2\,e^6-\frac {160\,B\,a^2\,b^3\,d\,e^5}{3}+20\,A\,a^2\,b^3\,e^6+\frac {160\,B\,a\,b^4\,d^2\,e^4}{3}-\frac {80\,A\,a\,b^4\,d\,e^5}{3}-\frac {128\,B\,b^5\,d^3\,e^3}{7}+\frac {32\,A\,b^5\,d^2\,e^4}{3}\right )}{b\,e^9}-\frac {x^2\,\left (10\,B\,a^4\,b\,e^6-120\,B\,a^3\,b^2\,d\,e^5+20\,A\,a^3\,b^2\,e^6+320\,B\,a^2\,b^3\,d^2\,e^4-120\,A\,a^2\,b^3\,d\,e^5-320\,B\,a\,b^4\,d^3\,e^3+160\,A\,a\,b^4\,d^2\,e^4+\frac {768\,B\,b^5\,d^4\,e^2}{7}-64\,A\,b^5\,d^3\,e^3\right )}{b\,e^9}-\frac {\frac {4\,B\,a^5\,d\,e^5}{15}+\frac {2\,A\,a^5\,e^6}{5}+\frac {16\,B\,a^4\,b\,d^2\,e^4}{3}+\frac {4\,A\,a^4\,b\,d\,e^5}{3}-64\,B\,a^3\,b^2\,d^3\,e^3+\frac {32\,A\,a^3\,b^2\,d^2\,e^4}{3}+\frac {512\,B\,a^2\,b^3\,d^4\,e^2}{3}-64\,A\,a^2\,b^3\,d^3\,e^3-\frac {512\,B\,a\,b^4\,d^5\,e}{3}+\frac {256\,A\,a\,b^4\,d^4\,e^2}{3}+\frac {2048\,B\,b^5\,d^6}{35}-\frac {512\,A\,b^5\,d^5\,e}{15}}{b\,e^9}+\frac {b^3\,x^5\,\left (\frac {2\,A\,b\,e}{5}+2\,B\,a\,e-\frac {24\,B\,b\,d}{35}\right )}{e^4}-\frac {x\,\left (70\,B\,a^5\,e^6+1400\,B\,a^4\,b\,d\,e^5+350\,A\,a^4\,b\,e^6-16800\,B\,a^3\,b^2\,d^2\,e^4+2800\,A\,a^3\,b^2\,d\,e^5+44800\,B\,a^2\,b^3\,d^3\,e^3-16800\,A\,a^2\,b^3\,d^2\,e^4-44800\,B\,a\,b^4\,d^4\,e^2+22400\,A\,a\,b^4\,d^3\,e^3+15360\,B\,b^5\,d^5\,e-8960\,A\,b^5\,d^4\,e^2\right )}{105\,b\,e^9}+\frac {b^2\,x^4\,\left (\frac {20\,B\,a^2\,e^2}{3}-\frac {20\,B\,a\,b\,d\,e}{3}+\frac {10\,A\,a\,b\,e^2}{3}+\frac {16\,B\,b^2\,d^2}{7}-\frac {4\,A\,b^2\,d\,e}{3}\right )}{e^5}+\frac {2\,B\,b^4\,x^6}{7\,e^3}\right )}{x^3\,\sqrt {d+e\,x}+\frac {a\,d^2\,\sqrt {d+e\,x}}{b\,e^2}+\frac {x^2\,\left (a\,e^9+2\,b\,d\,e^8\right )\,\sqrt {d+e\,x}}{b\,e^9}+\frac {d\,x\,\left (2\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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